6.15.28 problem 24

Internal problem ID [2026]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 24
Date solved : Tuesday, September 30, 2025 at 05:22:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5+4 x \right ) y^{\prime }+\left (9+4 x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 56

Order:=8; 
ode:=x^2*(1-2*x)*diff(diff(y(x),x),x)-x*(5+4*x)*diff(y(x),x)+(9+4*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (\left (-26\right ) x -324 x^{2}-\frac {6968}{3} x^{3}-\frac {37780}{3} x^{4}-57360 x^{5}-\frac {694736}{3} x^{6}-\frac {2566144}{3} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 +\left (1+20 x +180 x^{2}+1120 x^{3}+5600 x^{4}+24192 x^{5}+94080 x^{6}+337920 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )\right ) x^{3} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 136

ode=x^2*(1-2*x)*D[y[x],{x,2}]-x*(5+4*x)*D[y[x],x]+(9+4*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (337920 x^7+94080 x^6+24192 x^5+5600 x^4+1120 x^3+180 x^2+20 x+1\right ) x^3+c_2 \left (\left (-\frac {2566144 x^7}{3}-\frac {694736 x^6}{3}-57360 x^5-\frac {37780 x^4}{3}-\frac {6968 x^3}{3}-324 x^2-26 x\right ) x^3+\left (337920 x^7+94080 x^6+24192 x^5+5600 x^4+1120 x^3+180 x^2+20 x+1\right ) x^3 \log (x)\right ) \]

✓ Sympy. Time used: 0.453 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 2*x)*Derivative(y(x), (x, 2)) - x*(4*x + 5)*Derivative(y(x), x) + (4*x + 9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x^{3} + O\left (x^{8}\right ) \]